A family of multiplierless transforms is presented that approximate the eight-point type-II discrete cosine transform (DCT) as accurately as the state-of-the-art scaled DCT schemes, but having 14–17% shorter critical paths (1/6 or 1/7 less adders). Compared to the existing solutions that use the coordinate rotation digital computer (CORDIC) algorithm, the advantage of higher throughput is accompanied by saving additions. Only some lifting-based BinDCT schemes require less adders in total, in spite of longer critical paths. The transforms have been derived from the fast Loeffler's algorithm by replacing the rotation stage with unfolded CORDIC iterations, which have been arranged so that two rotation approximations use the same scaling. This is equivalent to imposing structural orthogonality (losslessness) on a system, from which the scaling can then be extracted so as to shorten the critical path. Supporting ideas are a notation for more conveniently describing CORDIC circuits, and an angle conversion that allows rotations to be approximated using an extended set of CORDIC circuits. The research results have been validated by field programmable gate array-based hardware design experiments and by usability tests based on a software JPEG codec.